Fractional weierstrass model for rough ocean surface and analytical derivation of its scattered field in a closed form

Analytical derivation of scattered field from rough ocean surface can be accomplished by a two-step analysis involving ocean surface modeling and scattered-field calculation. In the first step, the simplest linear model is constructed by assuming a rough ocean state as composition of infinite number of linear waves with identical crests and troughs. Unfortunately, regardless of ocean circumstances, ocean surface waves always exceed the linear range and yield a typical nonlinear asymmetric waveform with sharper peaks and shallow troughs. In this paper, by using curve-fit method and the concept of fractal geometry, we propose a new fractal function named fractional Weierstrass function which may be viewed as a generalization of classical Weierstrass function, and then, a fractional Weierstrass model for ocean surface is proposed by combining the fractional Weierstrass function with the Pierson-Moskowitz spectrum. An important advantage of our proposed model is the ability to represent nonlinearity with different degrees of ocean surface depending on a new parameter c. Some simulations demonstrate ratio c as an invariant scale of fractal ocean surface. The second step is strongly related to the surface modeling result of the first step. Within Kirchhoff approximation, an analytical derivation of scattered field in a closed form is obtained from our proposed fractional Weierstrass model surface. Moreover, some related discussions, including the normalization constant of the model, the correlation length, the scattered-field structure, and the influence of fractal and electromagnetic parameters over scattered field, are given. Finally, edge effects caused by the truncation of scatterer surface are also analyzed in detail.